Quantum approximate chaos optimization for use in a hybrid computing system

ABSTRACT

Embodiments described herein are generally related to a method and a system for performing a computation using a hybrid quantum-classical computing system, and, more specifically, to providing an approximate solution to a combinatorial optimization problem using a hybrid quantum-classical computing system that includes a group of trapped ions. A hybrid quantum-classical computing system that is able to provide a solution to a combinatorial optimization problem may include a classical computer, a system controller, and a quantum processor. The methods and systems described herein include an efficient method for an optimization routine executed by the classical computer in solving a problem in a hybrid quantum-classical computing system, which can provide improvement over the conventional method for an optimization by conventional stochastic optimization methods.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit to U.S. Provisional Application No.62/890,413, filed Aug. 22, 2019, which is incorporated by referenceherein.

BACKGROUND Field

The present disclosure generally relates to a method of performingcomputation in a hybrid quantum-classical computing system, and morespecifically, to a method of solving a combinatorial optimizationproblem in a hybrid computing system that includes a classical computerand quantum computer that includes a group of trapped ions.

Description of the Related Art

In quantum computing, quantum bits or qubits, which are analogous tobits representing a “0” and a “1” in a classical (digital) computer, arerequired to be prepared, manipulated, and measured (read-out) with nearperfect control during a computation process. Imperfect control of thequbits leads to errors that can accumulate over the computation process,limiting the size of a quantum computer that can perform reliablecomputations.

Among physical systems upon which it is proposed to build large-scalequantum computers, is a group of ions (e.g., charged atoms), which aretrapped and suspended in vacuum by electromagnetic fields. The ions haveinternal hyperfine states which are separated by frequencies in theseveral GHz range and can be used as the computational states of a qubit(referred to as “qubit states”). These hyperfine states can becontrolled using radiation provided from a laser, or sometimes referredto herein as the interaction with laser beams. The ions can be cooled tonear their motional ground states using such laser interactions. Theions can also be optically pumped to one of the two hyperfine stateswith high accuracy (preparation of qubits), manipulated between the twohyperfine states (single-qubit gate operations) by laser beams, andtheir internal hyperfine states detected by fluorescence uponapplication of a resonant laser beam (read-out of qubits). A pair ofions can be controllably entangled (two-qubit gate operations) byqubit-state dependent force using laser pulses that couple the ions tothe collective motional modes of a group of trapped ions, which arisefrom their Coulombic interaction between the ions. In general,entanglement occurs when pairs or groups of ions (or particles) aregenerated, interact, or share spatial proximity in ways such that thequantum state of each ion cannot be described independently of thequantum state of the others, even when the ions are separated by a largedistance.

In current state-of-the-art quantum computers, control of qubits isimperfect (noisy) and the number of qubits used in these quantumcomputers generally range from a hundred qubits to thousands of qubits.The number of quantum gates that can be used in such a quantum computer(referred to as a “noisy intermediate-scale quantum device” or “NISQdevice”) to construct circuits to run an algorithm within a controllederror rate is limited due to the noise.

For solving some optimization problems, a NISQ device having shallowcircuits can be used in combination with a classical computer (referredto as a hybrid quantum-classical computing system). In particular, infinding an approximate solution to combinatorial optimization problems,a quantum subroutine, which is run on a NISQ device, can be run as partof a classical optimization routine, which is run on a classicalcomputer. The classical computer (also referred to as a “classicaloptimizer”) instructs a controller to prepare the NISQ device (alsoreferred to as a “quantum processor”) in an N-qubit state, executequantum gate operations, and measure an outcome of the quantumprocessor. Subsequently, the classical optimizer instructs thecontroller to prepare the quantum processor in a slightly differentN-qubit state, and repeats execution of the gate operation andmeasurement of the outcome. This cycle is repeated until the approximatesolution can be extracted. Such hybrid quantum-classical computingsystem having a NISQ device may outperform classical computers infinding approximate solutions to such combinatorial optimizationproblems. However, time and resource required for the classicaloptimization routine based on conventional stochastic optimizationmethods increases exponentially as the targeted accuracy increases. Dueto inefficiencies in these conventional methods cause the outcomes ofthese processes to be too slow to be useful and too resource intensive.

Therefore, there is a need for an efficient method for the classicaloptimization routine in a hybrid quantum-classical computing system.

SUMMARY

Embodiments of the present disclosure provide a method of performingcomputation in a hybrid quantum-classical computing system including aclassical computer and a quantum processor. The method includesselecting, by a classical computer, a problem to be solved and computinga model Hamiltonian onto which the selected problem is mapped,selecting, by the classical computer, a set of variational parameters,setting a quantum processor in an initial state, where the quantumprocessor includes a group of trapped ions, each of which has twofrequency-separated states, transforming the quantum processor from theinitial state to a trial state based on the computed model Hamiltonianand the selected set of variational parameters, measuring a populationof the two frequency-separated states of each trapped ion in the quantumprocessor, and determining if a difference between the measuredpopulation and a previously measured population of the twofrequency-separated states of each trapped ion in the quantum processoris more or less than a predetermined value. The classical computereither selects another set of variational parameters based on a chaoticmap if it is determined that the difference is more than thepredetermined value and then sets the quantum processor in the initialstate, transforms the quantum processor from the initial state to a newtrial state, and measures a population of the two frequency-separatedstates of each trapped ion in the quantum processor after transformingthe quantum processor to the new trial state, or outputting the measuredpopulation as an optimized solution to the selected problem if it isdetermined that the difference is less than the predetermined value.Embodiments of the disclosure may also provide a hybridquantum-classical computing system comprising non-volatile memory havinga number of instructions stored therein which, when executed by one ormore processors, causes the hybrid quantum-classical computing system toperform operations of the method described above, and also other methodsdescribed herein.

Embodiments of the present disclosure also provide a hybridquantum-classical computing system. The hybrid quantum-classicalcomputing system includes a quantum processor comprising a group oftrapped ions, each of the trapped ions having two hyperfine statesdefining a qubit, one or more lasers configured to emit a laser beam,which is provided to trapped ions in the quantum processor, a classicalcomputer configured to select a problem to be solved, compute a modelHamiltonian onto which the selected problem is mapped, and select a setof variational parameters, and a system controller configured to set thequantum processor in an initial state, transform the quantum processorfrom the initial state to a trial state based on the computed modelHamiltonian and the selected set of variational parameters, and measurean expectation value of the model Hamiltonian on the quantum processor.The classical computer is further configured to determine if adifference between the measure population and a previously measuredpopulation of the two frequency-separated states of each trapped ion inthe quantum processor is less than a predetermined value. The classicalcomputer selects another set of variational parameters based on achaotic map if it is determined that the difference is more than thepredetermined value and then sets the quantum processor in the initialstate, transforms the quantum processor from the initial state to a newtrial state, measures an expectation value of the model Hamiltonian onthe quantum processor after transforming the quantum processor to thenew trial state, or output the measured expectation value of the modelHamiltonian as an optimized solution to the selected problem if it isdetermined that the difference is less than the predetermined value.Embodiments of the disclosure may also provide a hybridquantum-classical computing system comprising non-volatile memory havinga number of instructions stored therein which, when executed by one ormore processors, causes the hybrid quantum-classical computing system toperform operations of the method described above.

Embodiments of the present disclosure also provide a hybridquantum-classical computing system comprising non-volatile memory havinga number of instructions stored therein. The number of instructions,when executed by one or more processors, causes the hybridquantum-classical computing system to perform operations includingselect a problem to be solved and computing a model Hamiltonian ontowhich the selected problem is mapped, select a set of variationalparameters, set a quantum processor in an initial state, wherein thequantum processor comprises a plurality of qubits, transform the quantumprocessor from the initial state to a trial state based on the computedmodel Hamiltonian and the selected set of variational parameters,measure an expectation value of the model Hamiltonian on the quantumprocessor, and determine if a difference between the measuredexpectation value of the model Hamiltonian is more or less than apredetermined value. The computer program instructions further cause theinformation processing system to either select another set ofvariational parameters based on a chaotic map if it is determined thatthe difference is more than the predetermined value and then set thequantum processor in the initial state, transform the quantum processorfrom the initial state to a new trial state, and measure an expectationvalue of the model Hamiltonian on the quantum processor aftertransforming the quantum processor to the new trial state; or output themeasured population as an optimized solution to the selected problem ifit is determined that the difference is less than the predeterminedvalue. Embodiments of the disclosure may also provide a hybridquantum-classical computing system comprising non-volatile memory havinga number of instructions stored therein which, when executed by one ormore processors, causes the hybrid quantum-classical computing system toperform operations of the method described above.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above-recited features of the presentdisclosure can be understood in detail, a more particular description ofthe disclosure, briefly summarized above, may be had by reference toembodiments, some of which are illustrated in the appended drawings. Itis to be noted, however, that the appended drawings illustrate onlytypical embodiments of this disclosure and are therefore not to beconsidered limiting of its scope, for the disclosure may admit to otherequally effective embodiments.

FIG. 1 is a schematic partial view of an ion trap quantum computingsystem according to one embodiment.

FIG. 2 depicts a schematic view of an ion trap for confining ions in agroup according to one embodiment.

FIG. 3 depicts a schematic energy diagram of each ion in a group oftrapped ions according to one embodiment.

FIG. 4 depicts a qubit state of an ion represented as a point on asurface of the Bloch sphere.

FIGS. 5A, 5B, and 5C depict a few schematic collective transversemotional mode structures of a group of five trapped ions.

FIGS. 6A and 6B depict schematic views of motional sideband spectrum ofeach ion and a motional mode according to one embodiment.

FIG. 7 depicts an overall hybrid quantum-classical computing system forobtaining a solution to a combinatorial optimization problem by QuantumApproximate Optimization Algorithm (QAOA) according to one embodiment.

FIG. 8 depicts a flowchart illustrating a method of obtaining a solutionto a combinatorial optimization problem by Quantum ApproximateOptimization Algorithm (QAOA) according to one embodiment.

To facilitate understanding, identical reference numerals have beenused, where possible, to designate identical elements that are common tothe figures. In the figures and the following description, an orthogonalcoordinate system including an X-axis, a Y-axis, and a Z-axis is used.The directions represented by the arrows in the drawing are assumed tobe positive directions for convenience. It is contemplated that elementsdisclosed in some embodiments may be beneficially utilized on otherimplementations without specific recitation.

DETAILED DESCRIPTION

Embodiments described herein are generally related to a method and asystem for performing a computation using a hybrid quantum-classicalcomputing system, and, more specifically, to providing an approximatesolution to a combinatorial optimization problem using a hybridquantum-classical computing system that includes a group of trappedions.

A hybrid quantum-classical computing system that is able to provide asolution to a combinatorial optimization problem may include a classicalcomputer, a system controller, and a quantum processor. The classicalcomputer performs supporting and system control tasks includingselecting a combinatorial optimization problem to be run by use of auser interface, running a classical optimization routine, translatingthe series of logic gates into pulses to apply on the quantum processor,and pre-calculating parameters that optimize the pulses by use of acentral processing unit (CPU). A software program for performing thetasks is stored in a non-volatile memory within the classical computer.

The quantum processor includes trapped ions that are coupled withvarious hardware, including lasers to manipulate internal hyperfinestates (qubit states) of the trapped ions and an acousto-optic modulatorto read-out the internal hyperfine states (qubit states) of the trappedions. The system controller receives from the classical computerinstructions for controlling the quantum processor, controls varioushardware associated with controlling any and all aspects used to run theinstructions for controlling the quantum processor, and returns aread-out of the quantum processor and thus output of results of thecomputation(s) to the classical computer.

The methods and systems described herein include an efficient method foran optimization routine executed by the classical computer in solving aproblem in a hybrid quantum-classical computing system, which canprovide improvement over the conventional method for an optimization byconventional stochastic optimization methods.

General Hardware Configurations

FIG. 1 is a schematic partial view of an ion trap quantum computingsystem, or system 100, according to one embodiment. The system 100includes a classical (digital) computer 102, a system controller 104 anda quantum processor that is a group 106 of trapped ions (i.e., fiveshown) that extend along the Z-axis. The classical computer 102 includesa central processing unit (CPU), memory, and support circuits (or I/O).The memory is connected to the CPU, and may be one or more of a readilyavailable memory, such as a read-only memory (ROM), a random accessmemory (RAM), floppy disk, hard disk, or any other form of digitalstorage, local or remote. Software instructions, algorithms and data canbe coded and stored within the memory for instructing the CPU. Thesupport circuits (not shown) are also connected to the CPU forsupporting the processor in a conventional manner. The support circuitsmay include conventional cache, power supplies, clock circuits,input/output circuitry, subsystems, and the like.

An imaging objective 108, such as an objective lens with a numericalaperture (NA), for example, of 0.37, collects fluorescence along theY-axis from the ions and maps each ion onto a multi-channelphoto-multiplier tube (PMT) 110 for measurement of individual ions.Non-copropagating Raman laser beams from a laser 112, which are providedalong the X-axis, perform operations on the ions. A diffractive beamsplitter 114 creates an array of static Raman beams 116 that areindividually switched using a multi-channel acousto-optic modulator(AOM) 118 and is configured to selectively act on individual ions. Aglobal Raman laser beam 120 illuminates all ions at once. The systemcontroller (also referred to as a “RF controller”) 104 controls the AOM118 and thus controls laser pulses to be applied to trapped ions in thegroup 106 of trapped ions. The system controller 104 includes a centralprocessing unit (CPU) 122, a read-only memory (ROM) 124, a random accessmemory (RAM) 126, a storage unit 128, and the like. The CPU 122 is aprocessor of the system controller 104. The ROM 124 stores variousprograms and the RAM 126 is the working memory for various programs anddata. The storage unit 128 includes a nonvolatile memory, such as a harddisk drive (HDD) or a flash memory, and stores various programs even ifpower is turned off. The CPU 122, the ROM 124, the RAM 126, and thestorage unit 128 are interconnected via a bus 130. The system controller104 executes a control program which is stored in the ROM 124 or thestorage unit 128 and uses the RAM 126 as a working area. The controlprogram will include software applications that include program codethat may be executed by processor in order to perform variousfunctionalities associated with receiving and analyzing data andcontrolling any and all aspects of the methods and hardware used tocreate the ion trap quantum computer system 100 discussed herein.

FIG. 2 depicts a schematic view of an ion trap 200 (also referred to asa Paul trap) for confining ions in the group 106 according to oneembodiment. The confining potential is exerted by both static (DC)voltage and radio frequency (RF) voltages. A static (DC) voltage V_(S)is applied to end-cap electrodes 210 and 212 to confine the ions alongthe Z-axis (also referred to as an “axial direction” or a “longitudinaldirection”). The ions in the group 106 are nearly evenly distributed inthe axial direction due to the Coulomb interaction between the ions. Insome embodiments, the ion trap 200 includes four hyperbolically-shapedelectrodes 202, 204, 206, and 208 extending along the Z-axis.

During operation, a sinusoidal voltage V₁ (with an amplitude V_(RF)/2)is applied to an opposing pair of the electrodes 202, 204 and asinusoidal voltage V₂ with a phase shift of 180° from the sinusoidalvoltage V₁ (and the amplitude V_(RF)/2) is applied to the other opposingpair of the electrodes 206, 208 at a driving frequency ω_(RF),generating a quadrupole potential. In some embodiments, a sinusoidalvoltage is only applied to one opposing pair of the electrodes 202, 204,and the other opposing pair 206, 208 is grounded. The quadrupolepotential creates an effective confining force in the X-Y planeperpendicular to the Z-axis (also referred to as a “radial direction” or“transverse direction”) for each of the trapped ions, which isproportional to a distance from a saddle point (i.e., a position in theaxial direction (Z-direction)) at which the RF electric field vanishes.The motion in the radial direction (i.e., direction in the X-Y plane) ofeach ion is approximated as a harmonic oscillation (referred to assecular motion) with a restoring force towards the saddle point in theradial direction and can be modeled by spring constants k_(x) and k_(y),respectively, as is discussed in greater detail below. In someembodiments, the spring constants in the radial direction are modeled asequal when the quadrupole potential is symmetric in the radialdirection. However, undesirably in some cases, the motion of the ions inthe radial direction may be distorted due to some asymmetry in thephysical trap configuration, a small DC patch potential due toinhomogeneity of a surface of the electrodes, or the like and due tothese and other external sources of distortion the ions may lieoff-center from the saddle points.

FIG. 3 depicts a schematic energy diagram 300 of each ion in the group106 of trapped ions according to one embodiment. In one example, eachion may be a positive Ytterbium ion, ¹⁷¹Yb⁺, which has the ²S_(1/2)hyperfine states (i.e., two electronic states) with an energy splitcorresponding to a frequency difference (referred to as a “carrierfrequency”) of ω₀₁/2π=12.642821 GHz. A qubit is formed with the twohyperfine states, denoted as |0

and |1

, where the hyperfine ground state (i.e., the lower energy state of the²S_(1/2) hyperfine states) is chosen to represent |0

. Hereinafter, the terms “hyperfine states,” “internal hyperfinestates,” and “qubits” may be interchangeably used to represent |0

and |1

. Each ion may be cooled (i.e., kinetic energy of the ion may bereduced) to near the motional ground state |0

_(m) for any motional mode m with no phonon excitation (i.e., n_(ph)=0)by known laser cooling methods, such as Doppler cooling or resolvedsideband cooling, and then the qubit state prepared in the hyperfineground state |0

by optical pumping. Here, |0

represents the individual qubit state of a trapped ion whereas |0

_(m) with the subscript m denotes the motional ground state for amotional mode m of a group 106 of trapped ions.

An individual qubit state of each trapped ion may be manipulated by, forexample, a mode-locked laser at 355 nanometers (nm) via the excited²P_(1/2) level (denoted as |e

). As shown in FIG. 3, a laser beam from the laser may be split into apair of non-copropagating laser beams (a first laser beam with frequencyω₁ and a second laser beam with frequency ω₂) in the Ramanconfiguration, and detuned by a one-photon transition detuning frequencyΔ=ω₁−ω_(0e) with respect to the transition frequency ω_(0e) between |0

and |e

, as illustrated in FIG. 3. A two-photon transition detuning frequency δincludes adjusting the amount of energy that is provided to the trappedion by the first and second laser beams, which when combined is used tocause the trapped ion to transfer between the hyperfine states |0

and |1

. When the one-photon transition detuning frequency Δ is much largerthan a two-photon transition detuning frequency (also referred to simplyas “detuning frequency”) δ=ω₁−ω₂−ω₀₁ (hereinafter denoted as ±μ, μ beinga positive value), single-photon Rabi frequencies Ω_(0e)(t) andΩ_(1e)(t) (which are time-dependent, and are determined by amplitudesand phases of the first and second laser beams), at which Rabi floppingbetween states |0

and |e

and between states |1

and |e

respectively occur, and a spontaneous emission rate from the excitedstate |e

, Rabi flopping between the two hyperfine states |0

and |1

(referred to as a “carrier transition”) is induced at the two-photonRabi frequency Ω(t). The two-photon Rabi frequency Ω(t) has an intensity(i.e., absolute value of amplitude) that is proportional toΩ_(0e)Ω_(1e)/2Δ, where Ω_(0e) and Ω_(1e) are the single-photon Rabifrequencies due to the first and second laser beams, respectively.Hereinafter, this set of counter-propagating laser beams in the Ramanconfiguration to manipulate internal hyperfine states of qubits (qubitstates) may be referred to as a “composite pulse” or simply as a“pulse,” and the resulting time-dependent pattern of the two-photon Rabifrequency Ω(t) may be referred to as an “amplitude” of a pulse or simplyas a “pulse,” which are illustrated and further described below. Thedetuning frequency δ=ω₁−ω₂−ω₀₁ may be referred to as detuning frequencyof the composite pulse or detuning frequency of the pulse. The amplitudeof the two-photon Rabi frequency Ω(t), which is determined by amplitudesof the first and second laser beams, may be referred to as an“amplitude” of the composite pulse.

It should be noted that the particular atomic species used in thediscussion provided herein is just one example of atomic species whichhas stable and well-defined two-level energy structures when ionized andan excited state that is optically accessible, and thus is not intendedto limit the possible configurations, specifications, or the like of anion trap quantum computer according to the present disclosure. Forexample, other ion species include alkaline earth metal ions (Be⁺, Ca⁺,Sr⁺, Mg⁺, and Ba⁺) or transition metal ions (Zn⁺, Hg⁺, Cd⁺).

FIG. 4 is provided to help visualize a qubit state of an ion isrepresented as a point on a surface of the Bloch sphere 400 with anazimuthal angle ϕ and a polar angle θ. Application of the compositepulse as described above, causes Rabi flopping between the qubit state|0

(represented as the north pole of the Bloch sphere) and |1

(the south pole of the Bloch sphere) to occur. Adjusting time durationand amplitudes of the composite pulse flips the qubit state from |0

to |1

(i.e., from the north pole to the south pole of the Bloch sphere), orthe qubit state from |1

to |0

(i.e., from the south pole to the north pole of the Bloch sphere). Thisapplication of the composite pulse is referred to as a “π-pulse”.Further, by adjusting time duration and amplitudes of the compositepulse, the qubit state |0

may be transformed to a superposition state |0

+|1

, where the two qubit states |0

and |1

are added and equally-weighted in-phase (a normalization factor of thesuperposition state is omitted hereinafter without loss of generality)and the qubit state |1

to a superposition state |0

−|1

, where the two qubit states |0

and |1

are added equally-weighted but out of phase. This application of thecomposite pulse is referred to as a “π/2-pulse”. More generally, asuperposition of the two qubits states |0

and |1

that are added and equally-weighted is represented by a point that lieson the equator of the Bloch sphere. For example, the superpositionstates |0

±|1

correspond to points on the equator with the azimuthal angle ϕ beingzero and π, respectively. The superposition states that correspond topoints on the equator with the azimuthal angle ϕ are denoted as |0

+e^(iϕ)|1

(e.g., |0

±|1

for ϕ±π/2). Transformation between two points on the equator (i.e., arotation about the Z-axis on the Bloch sphere) can be implemented byshifting phases of the composite pulse.

Entanglement Formation

FIGS. 5A, 5B, and 5C depict a few schematic structures of collectivetransverse motional modes (also referred to simply as “motional modestructures”) of a group 106 of five trapped ions, for example. Here, theconfining potential due to a static voltage V_(S) applied to the end-capelectrodes 210 and 212 is weaker compared to the confining potential inthe radial direction. The collective motional modes of the group 106 oftrapped ions in the transverse direction are determined by the Coulombinteraction between the trapped ions combined with the confiningpotentials generated by the ion trap 200. The trapped ions undergocollective transversal motions (referred to as “collective transversemotional modes,” “collective motional modes,” or simply “motionalmodes”), where each mode has a distinct energy (or equivalently, afrequency) associated with it. A motional mode having the m-th lowestenergy is hereinafter referred to as |n_(ph)

_(m), where n_(ph) denotes the number of motional quanta (in units ofenergy excitation, referred to as phonons) in the motional mode, and thenumber of motional modes M in a given transverse direction is equal tothe number of trapped ions N in the group 106. FIGS. 5A-5C schematicallyillustrates examples of different types of collective transversemotional modes that may be experienced by five trapped ions that arepositioned in a group 106. FIG. 5A is a schematic view of a commonmotional mode |n_(ph)

_(M) having the highest energy, where M is the number of motional modes.In the common motional mode |n

_(M), all ions oscillate in phase in the transverse direction. FIG. 5Bis a schematic view of a tilt motional mode |n_(ph)

_(M−1) which has the second highest energy. In the tilt motional mode,ions on opposite ends move out of phase in the transverse direction(i.e., in opposite directions). FIG. 5C is a schematic view of ahigher-order motional mode |n_(ph)

_(M−3) which has a lower energy than that of the tilt motional mode|n_(ph)

_(M−1), and in which the ions move in a more complicated mode pattern.

It should be noted that the particular configuration described above isjust one among several possible examples of a trap for confining ionsaccording to the present disclosure and does not limit the possibleconfigurations, specifications, or the like of traps according to thepresent disclosure. For example, the geometry of the electrodes is notlimited to the hyperbolic electrodes described above. In other examples,a trap that generates an effective electric field causing the motion ofthe ions in the radial direction as harmonic oscillations may be amulti-layer trap in which several electrode layers are stacked and an RFvoltage is applied to two diagonally opposite electrodes, or a surfacetrap in which all electrodes are located in a single plane on a chip.Furthermore, a trap may be divided into multiple segments, adjacentpairs of which may be linked by shuttling one or more ions, or coupledby photon interconnects. A trap may also be an array of individualtrapping regions arranged closely to each other on a micro-fabricatedion trap chip. In some embodiments, the quadrupole potential has aspatially varying DC component in addition to the RF component describedabove.

In an ion trap quantum computer, the motional modes may act as a databus to mediate entanglement between two qubits and this entanglement isused to perform an XX gate operation. That is, each of the two qubits isentangled with the motional modes, and then the entanglement istransferred to an entanglement between the two qubits by using motionalsideband excitations, as described below. FIGS. 6A and 6B schematicallydepict views of a motional sideband spectrum for an ion in the group 106in a motional mode |n_(ph)

_(M) having frequency om according to one embodiment. As illustrated inFIG. 6B, when the detuning frequency of the composite pulse is zero(i.e., a frequency difference between the first and second laser beamsis tuned to the carrier frequency, δ=ω₁−ω₂−ω₀₁=0), simple Rabi floppingbetween the qubit states |0

and |1

(carrier transition) occurs. When the detuning frequency of thecomposite pulse is positive (i.e., the frequency difference between thefirst and second laser beams is tuned higher than the carrier frequency,δ=ω₁−ω₂−ω₀₁=μ>0, referred to as a blue sideband), Rabi flopping betweencombined qubit-motional states |0

|n_(ph)

_(m) and |1

|n_(ph)+1

_(m) occurs (i.e., a transition from the m-th motional mode withn-phonon excitations denoted by |n_(ph)

_(m) to the m-th motional mode with (n_(ph)+1)-phonon excitationsdenoted by |n_(ph)+1

_(m) occurs when the qubit state |0

flips to |1

). When the detuning frequency of the composite pulse is negative (i.e.,the frequency difference between the first and second laser beams istuned lower than the carrier frequency by the frequency ω_(m) of themotional mode |n_(ph)

_(m), δ=ω₁−ω₂−ω₀₁=−μ<0, referred to as a red sideband), Rabi floppingbetween combined qubit-motional states |0

|n_(ph)

_(m) and |1

|n_(ph)−|1

_(m) occurs (i.e., a transition from the motional mode |n_(ph)

_(m) to the motional mode |n_(ph)−|1

_(m) with one less phonon excitations occurs when the qubit state |0

flips to |1

). A π/2-pulse on the blue sideband applied to a qubit transforms thecombined qubit-motional state |0

|n_(ph)

_(m) into a superposition of |0

|n_(ph))_(m) and |1

|_(ph)+1

_(m). A π/2-pulse on the red sideband applied to a qubit transforms thecombined qubit-motional |0

n_(ph)

_(m) into a superposition of |0

|n_(ph)

_(m) and |1

|n_(ph)−|1

_(m). When the two-photon Rabi frequency Ω(t) is smaller as compared tothe detuning frequency δ=ω₁−ω₂−ω₀₁=±μ, the blue sideband transition orthe red sideband transition may be selectively driven. Thus, qubitstates of a qubit can be entangled with a desired motional mode byapplying the right type of pulse, such as a π/2-pulse, which can besubsequently entangled with another qubit, leading to an entanglementbetween the two qubits that is needed to perform an XX-gate operation inan ion trap quantum computer.

By controlling and/or directing transformations of the combinedqubit-motional states as described above, an XX-gate operation may beperformed on two qubits (i-th and j-th qubits). In general, the XX-gateoperation (with maximal entanglement) respectively transforms two-qubitstates |0

_(i)|0

_(j), |0

_(i)|1

_(j), |1

_(i)|0

_(j), and as follows:

|0

_(i)|0

_(j)→|0

_(i)|0

_(j) −i|1

_(i)|1

_(j)

|0

_(i)|1

_(j)→|0

_(i)|1

_(j) −i|1

_(i)|0

_(j)

|1

_(i)|0

_(j) →−i|0

_(i)|1

_(j)+|1

_(i)|0

_(j)

|1

_(i)|1

_(j) →−i|0

_(i)|0

_(j)+|1

_(i)|1

_(j).

For example, when the two qubits (i-th and j-th qubits) are bothinitially in the hyperfine ground state |0

(denoted as |0

_(i)|0

_(j)) and subsequently a π/2-pulse on the blue sideband is applied tothe i-th qubit, the combined state of the i-th qubit and the motionalmode |0

_(i)|n_(ph)

_(m) is transformed into a superpositin of |0

_(i)|n_(ph)

_(m) and |1

_(i)n_(ph)+1

_(m), and thus the combined state of the two qubits and the motionalmode is transformed into a superposition of |0

_(i)|0

_(j)|n_(ph)

_(m) and |1

_(j)|n_(ph)+1

_(m). When a π/2-pulse on the red sideband is applied to the j-th qubit,the combined state of the j-th qubit and the motional mode |0

_(j)|n_(ph)

_(m) is transformed to superposition of |0

_(j)|n_(ph)

_(m) and |1

_(j)|n_(ph)−|1

_(m) and the combined state |0

_(j)|n_(ph)+1

_(m) is transformed into a superposition of |0

_(j)|n_(ph)+1

_(m) and |1

_(j)|n_(ph)

_(p).

Thus, applications of a π/2-pulse on the blue sideband on the i-th qubitand a π/2-pulse on the red sideband on the j-th qubit may transform thecombined state of the two qubits and the motional mode |0

_(i)|0

_(j)|n_(ph)

_(m) into a superposition of |0

_(i)|0

_(j)|n_(ph)

_(m) and |1

_(i)|1

_(j)|n_(ph)

_(m), the two qubits now being in an entangled state. For those ofordinary skill in the art, it should be clear that two-qubit states thatare entangled with motional mode having a different number of phononexcitations from the initial number of phonon excitations n_(ph) (i.e.,|1

_(i)|0

_(j)|n_(ph)+1

_(m) and |0

_(i)|1

_(j)|n_(ph)−|1

_(m)) can be removed by a sufficiently complex pulse sequence, and thusthe combined state of the two qubits and the motional mode after theXX-gate operation may be considered disentangled as the initial numberof phonon excitations n_(ph) in the m-th motional mode stays unchangedat the end of the XX-gate operation. Thus, qubit states before and afterthe XX-gate operation will be described below generally withoutincluding the motional modes.

More generally, the combined state of i-th and j-th qubits transformedby the application of pulses on the sidebands for duration τ (referredto as a “gate duration”), having amplitudes Ω^((i)) and Ω^((i)) anddetuning frequency μ, can be described in terms of an entanglinginteraction X^((i,j))(τ) as follows:

|0

_(i)|0

_(j)→cos(2χ^((i,j))(τ)|0

_(i)|0

_(j) −i sin(2χ^((i,j))(τ))|1

_(i)|1

_(j)

|0

_(i)|1

_(j)→cos(2χ^((i,j))(τ))|0

_(i)|1

_(j) −i sin(2χ^((i,j))(τ))|1

_(i)|0

_(j)

|1

_(i)|0

_(j) →−i sin(2χ^((i,j))(τ))|0

_(i)|1

_(nj)+cos(2χ^((i,j))(τ))|1

_(i)|0

_(j)

|1

_(i)|1

_(j) →−i sin(2χ^((i,j))(τ))|0

_(i)|0

_(j)+cos(2χ^((i,j))(τ))|1

_(i)|1

_(j)

where,

${\chi^{({i,j})}(\tau)} = {{- 4}{\sum\limits_{m = 1}^{M}{\eta_{m}^{(i)}\eta_{m}^{(j)}{\int\limits_{0}^{\tau}{dt_{2}{\int\limits_{0}^{t_{2}}{dt_{1}{\Omega^{(i)}( t_{2} )}{\Omega^{(j)}( t_{1} )}{\cos ( {\mu t_{2}} )}{\cos ( {\mu t_{1}} )}{\sin \lbrack {\omega_{m}( {t_{2} - t_{1}} )} \rbrack}}}}}}}}$

and η_(m) ^((i)) is the Lamb-Dicke parameter that quantifies thecoupling strength between the i-th ion and the m-th motional mode havingthe frequency ω_(m), and M is the number of the motional modes (equal tothe number N of ions in the group 106).

The entanglement interaction between two qubits described above can beused to perform an XX-gate operation. The XX-gate operation (XX gate)along with single-qubit operations (R gates) forms a set of gates {R,XX} that can be used to build a quantum computer that is configured toperform desired computational processes. Among several known sets oflogic gates by which any quantum algorithm can be decomposed, a set oflogic gates, commonly denoted as {R, XX}, is native to a quantumcomputing system of trapped ions described herein. Here, the R gatecorresponds to manipulation of individual qubit states of trapped ions,and the XX gate (also referred to as an “entangling gate”) correspondsto manipulation of the entanglement of two trapped ions.

To perform an XX-gate operation between i-th and j-th qubits, pulsesthat satisfy the condition χ^((i,j))(τ)=θ^((i,j)) (0<θ^((i,j))≤π/8)(i.e., the entangling interaction χ^((i,j))(τ) has a desired valueθ^((i,j)), referred to as condition for a non-zero entanglementinteraction) are constructed and applied to the i-th and the j-thqubits. The transformations of the combined state of the i-th and thej-th qubits described above corresponds to the XX-gate operation withmaximal entanglement when θ^((i,j))=π/8. Amplitudes Ω^((i))(t) andΩ^((j))(t) of the pulses to be applied to the i-th and the j-th qubitsare control parameters that can be adjusted to ensure a non-zero tunableentanglement of the i-th and the j-th qubits to perform a desired XXgate operation on i-th and j-th qubits.

Hybrid Quantum-Classical Computing System

While currently available quantum computers may be noisy and prone toerrors, a combination of both quantum and classical computers, in whicha quantum computer is a domain-specific accelerator, may be able tosolve optimization problems that are beyond the reach of classicalcomputers. An example of such optimization problems is in solvingcombinatorial optimization problems, where Quantum ApproximateOptimization Algorithm (QAOA) perform search for optimal solutions froma set of possible solutions according to some given criteria, using aquantum computer and a classical computer. The combinatorialoptimization problems that can be solved by the methods described hereinmay include the PageRank (PR) problem for ranking web pages in searchengine results and the maximum-cut (MaxCut) problem with applications inclustering, network science, and statistical physics. The MaxCut problemaims at grouping nodes of a graph into two partitions by cutting acrosslinks between them in such a way that a weighted sum of intersectededges is maximized. The combinatorial optimization problems that can besolved by the methods described herein may further include thetravelling salesman problem for finding shortest and/or cheapest roundtrips visiting all given cities. The travelling salesman problem isapplied to scheduling a printing press for a periodical withmulti-editions, scheduling school buses minimizing the number of routesand total distance while no bus is overloaded or exceeds a maximumallowed policy, scheduling a crew of messengers to pick up deposit frombranch banks and return the deposit to a central bank, determining anoptimal path for each army planner to accomplish the goals of themission in minimum possible time, designing global navigation satellitesystem (GNSS) surveying networks, and the like. Another combinatorialoptimization problem is the knapsack problem to find a way to pack aknapsack to get the maximum total value, given some items. The knapsackproblem is applied to resource allocation given financial constraints inhome energy management, network selection for mobile nodes, cognitiveradio networks, sensor selection in distributed multiple radar, or thelike.

A combinatorial optimization problem is modeled by an objective function(also referred to as a cost function) that maps events or values of oneor more variables onto real numbers representing “cost” associated withthe events or values and seeks to minimize the cost function. In somecases, the combinatorial optimization problem may seek to maximize theobjective function. The combinatorial optimization problem is furthermapped onto a simple physical system described by a model Hamiltonian(corresponding to the sum of kinetic energy and potential energy of allparticles in the system) and the problem seeks the low-lying energystate of the physical system.

This hybrid quantum-classical computing system has at least thefollowing advantages. First, an initial guess is derived from aclassical computer, and thus the initial guess does not need to beconstructed in a quantum processor that may not be reliable due toinherent and unwanted noise in the system. Second, a quantum processorperforms a small-sized (e.g., between a hundred qubits and a fewthousand qubits) but accelerated operation (that can be performed usinga small number of quantum logic gates) between an input of a guess fromthe classical computer and a measurement of a resulting state, and thusa NISQ device can execute the operation without a significant amount ofaccumulating errors. Thus, the hybrid quantum-classical computing systemmay allow challenging problems to be solved, such as small butchallenging combinatorial optimization problems, which are notpractically feasible on classical computers, or suggest ways to speed upthe computation with respect to the results that would be achieved usingthe best known classical algorithm.

FIGS. 7 and 8 depict an overall hybrid quantum-classical computingsystem 700 and a flowchart illustrating a method 800 of obtaining asolution to a combinatorial optimization problem by Quantum ApproximateOptimization Algorithm (QAOA) according to one embodiment. In thisexample, the quantum processor is the group 106 of N trapped ions, inwhich the two hyperfine states of each of the N trapped ions form aqubit.

The Approximate Optimization Algorithm (QAOA) relies on a variationalsearch by a well-known variational method. The variational methodconsists of iterations that include choosing a “trial state” of thequantum processor depending on a set of one or more parameters (referredto as “variational parameters”) and measuring an expectation value ofthe model Hamiltonian (e.g., energy) of the trial state. A set ofvariational parameters (and thus a corresponding trial state) isadjusted and an optimal set of variational parameters are found thatminimizes the expectation value of the model Hamiltonian (the energy).The resulting energy is an approximation to the exact lowest energystate.

In block 802, by the classical computer 102, a combinatorialoptimization problem to be solved is selected, for example, by use of auser interface of the classical computer 102, or retrieved from thememory of the classical computer 102, and a model Hamiltonian H_(C), towhich the selected combinatorial optimization problem is mapped, iscomputed.

In a combinatorial optimization problem defined on a set of N binaryvariables with t constrains (α=1, 2, . . . t), the objective function isthe number of satisfied clauses C(z)=Σ_(α=1) ^(t)C_(α) (z) or a weightedsum of satisfied clauses C(z)=τ_(α=1) ^(t)h_(α)C_(α) (z) (h_(α)corresponds to a weight for each constraint α), where z=z₁ z₂ . . .z_(N) is a N-bit string and C_(α)(z)=1 if z satisfies the constraint α.The clause C_(α)(z) that describes the constraint α typically includes asmall number of variables z_(i). The goal is to minimize the objectivefunction. Minimizing this objective function can be converted to findinga low-lying energy state of a model Hamiltonian H_(C)=τ_(α=1) ^(t)h_(α)P_(α) by mapping each binary variable z_(i) to a quantum spin σ_(i) ^(z)and the constraints to the couplings among the quantum spins σ_(i) ^(z),where P_(α) is a Pauli string (also referred to as a Pauli term)P_(α)=σ₁ ^(α) ¹ ⊗σ₂ ^(α) ² ⊗ . . . σ_(N) ^(α) ^(N) and σ_(N) ^(α) ^(i)is either the identity operator I or the Pauli matrix σ_(i) ^(X), σ_(i)^(Y), or σ_(i) ^(z). Here t stands for the number of couplings among thequantum spins and h_(α) (α=1, 2, . . . , t) stands for the strength ofthe coupling α.

The quantum processor 106 has N qubits and each quantum spin σ_(i) ^(z)(i=1, 2, . . . , N) is encoded in qubit i (i=1, 2, . . . , N) in thequantum processor 106. For example, the spin-up and spin-down states ofthe quantum spin σ_(i) ^(z) are encoded as |0

and |1

of the qubit i.

In block 804, following the mapping of the selected combinatorialoptimization problem onto a model Hamiltonian H_(C)=Σ_(α=1) ^(t)h_(α)P_(α), a set of variational parameters ({right arrow over (γ)}=γ₁, γ₂, .. . , γ_(p), {right arrow over (β)}=β₁, β₂, . . . , β_(p)) is selected,by the classical computer 102, to construct a sequence of gates (alsoreferred to a “trial state preparation circuit”) A({right arrow over(γ)}, {right arrow over (β)}), which prepares the quantum processor 106in a trial state |Ψ({right arrow over (γ)}, {right arrow over (β)})

. For the initial iteration, a set of variational parameters {rightarrow over (γ)}, {right arrow over (β)} may be randomly chosen. Thistrial state |Ψ({right arrow over (γ)}, {right arrow over (β)})

is used to provide an expectation value of each Pauli term P_(α) (α=1,2, . . . , t) of the model Hamiltonian H_(C). The trial statepreparation circuit A({right arrow over (γ)}, {right arrow over (β)})includes p layers (i.e., p-time repetitions) of a model-Hamiltoniancircuit U(γ_(l))=e^(−iγ) ^(l) ^(H) ^(C) and a mixing circuitU_(Mix)(β_(l)) that relates to a mixing term

$H_{B} = {\sum_{i = 1}^{n}{{\sigma_{i}^{{\overset{¯}{\alpha}}_{i}}( {{U_{Mix}( \beta_{l} )} = e^{{- i}\beta_{l}H_{B}}} )}( {{l = 1},2,\ldots \mspace{14mu},\ p} )}}$

as

A({right arrow over (γ)},{right arrow over (β)})=U_(Mix)(β_(p))U(γ_(p))U _(Mix)(β_(p−1))U(γ_(p−1)) . . . U_(Mix)(β₁)U(γ₁).

Each term

in the mixing term H_(B) corresponds to an orthogonal Pauli matrix toσ_(i) ^(α) ^(i) .

In block 806, following the selection of a set of variational parameters{right arrow over (γ)}, {right arrow over (β)}, the quantum processor106 is set in an initial state |Ψ₀

by the system controller 104. The initial state |Ψ₀

may be in the hyperfine ground state of the quantum processor 106. Aqubit can be set in the hyperfine ground state |0

by optical pumping and in the superposition state |0

+|1

by application of a proper combination of single-qubit operations(denoted by “H” in FIG. 7) to the hyperfine ground state |0

.

In block 808, following the preparation of the quantum processor 106 inthe initial state |Ψ₀

, the trial state preparation circuit A({right arrow over (γ)}, {rightarrow over (β)}) is applied to the quantum processor 106, by the systemcontroller 104, to construct the trial state |Ψ({right arrow over (γ)},{right arrow over (β)})

. The trial state preparation circuit A({right arrow over (γ)}, {rightarrow over (β)}) is decomposed into series of XX-gate operations (XXgates) and single-qubit operations (R gates) and optimized by theclassical computer 102. The series of XX-gate operations (XX gates) andsingle-qubit operations (R gates) can be implemented by application of aseries of laser pulses, intensities, durations, and detuning of whichare appropriately adjusted by the classical computer 102 on the setinitial state |Ψ₀

and transform the quantum processor from the initial state |Ψ₀

to trial state |Ψ({right arrow over (γ)}, {right arrow over (β)})

.

In block 810, following the construction of the trial state |Ψ({rightarrow over (γ)}, {right arrow over (β)})

on the quantum processor 106, the expectation value F_(α)({right arrowover (γ)}, {right arrow over (β)})=

Ψ({right arrow over (γ)}, {right arrow over (β)})|P_(α)|Ψ({right arrowover (γ)}, {right arrow over (β)})

of the Pauli term P_(α) (α=1, 2, . . . , m) is measured by the systemcontroller 104. In one embodiment, appropriate basis changes are made toindividual qubits for a given P_(α) prior to measurement so thatrepeated measurements of populations of the trapped ions in the group106 of trapped ions (by collecting fluorescence from each trapped ionand mapping onto the PMT 110) yield the expectation value the Pauli termP_(α) of the trial state |Ψ({right arrow over (γ)}, {right arrow over(β)})

.

In block 812, following the measurement of the expectation value of thePauli term P_(α) (α=1, 2, . . . , m), blocks 806 to 810 for anotherPauli term P_(α) (α=1, 2, . . . , m) until the expectation values of allthe Pauli terms P_(α) (α=1, 2, . . . , m) of the model HamiltonianH_(C)=Σ_(α=1) ^(m)h_(α) P_(α) have been measured by the systemcontroller 104.

In block 814, following the measurement of the expectation values of allthe Pauli terms P_(α) (α=1, 2, . . . , m), a sum of the measuredexpectation values of all the Pauli terms P_(α) (α=1, 2, . . . , m) ofthe model Hamiltonian H_(C)=Σ_(α=1) ^(m)h_(α) P_(α) (that is, themeasured expectation value of the model Hamiltonian H_(C), F({rightarrow over (γ)}, {right arrow over (β)})=Σ_(α=1) ^(m) F_(α)({right arrowover (γ)}, {right arrow over (β)})=Σ_(α=1) ^(m)

Ψ({right arrow over (γ)}, {right arrow over (β)})|P_(α)|Ψ({right arrowover (γ)}, {right arrow over (β)})

) is computed, by the classical computer 102.

In block 816, following the computation of the measured expectationvalue of the model Hamiltonian H_(C), the measured expectation valueF({right arrow over (γ)}, {right arrow over (β)}) of the modelHamiltonian H_(C) is compared to the measured expectation value of themodel Hamiltonian H_(C) in the previous iteration, by the classicalcomputer 102. If a difference between the two values is less than apredetermined value (i.e., the expectation value sufficiently convergestowards a fixed value), the method proceeds to block 820. If thedifference between the two values is more than the predetermined value,the method proceeds to block 818.

In block 818, another set of variational parameters {right arrow over(γ)}, {right arrow over (β)} for a next iteration of blocks 806 to 816is computed by the classical computer 102, in search for an optimal setof variational parameters {right arrow over (γ)}, {right arrow over (β)}to minimize the expectation value of the model Hamiltonian H_(C),F({right arrow over (γ)}, {right arrow over (β)})=Σ_(α=1) ^(m)

Ψ({right arrow over (γ)}, {right arrow over (β)})|P_(α)Ψ({right arrowover (γ)}, {right arrow over (β)})

. That is, the classical computer 102 will execute a classicaloptimization routine to find the optimal set of variational parameters{right arrow over (γ)}, {right arrow over (β)} (F({right arrow over(γ)}, {right arrow over (β)})). As the number of layers p increases, theaccuracy of the measured expectation value of the model HamiltonianH_(C) is improved. However, with increasing p, circuit depth of thetrial state preparation circuit A ({right arrow over (γ)}, {right arrowover (β)}) increases and errors due to the noise in a NISQ device willbe accumulated. Furthermore, a variational search space in which theoptimal variational parameters (the number of which is 2p) is searchedby a conventional classical stochastic optimization algorithm, such assimultaneous perturbation stochastic approximation (SPSA), particleswarm optimization (PSO), Bayesian optimization (BO), and Nelder-Mead(NM), generally increases exponentially and thus the optimizationbecomes generally exponentially difficult as the number of p increases.

In the embodiments described herein, a set of variational parameters{right arrow over (γ)}, {right arrow over (β)} for a next iteration iscomputed by replacing all or some of γ₁, γ₂, . . . , γ_(p) and β₁, β₂, .. . , β_(p) according to the functions ƒ(γ_(l), p, M, r, s) and g(β_(l),p, M, r, s), respectively, where the functions ƒ(γ_(l), p, M, r, s) andg(β_(l), p, M, r, s) are well-known chaotic maps that update given thevariational parameters γ_(l), β_(l), to the variational parametersγ′_(l), β′_(l), for a next iteration as γ′_(l)=ƒ(γ_(l), p, M, r, s) andβ′_(l)=g(β_(l), p, M, r, s), respectively, based on a chaotic map M.Here, r represents a tuning parameter, and s represents the number ofevaluations of the chaotic map M. In some embodiments, the processesperformed during block 818 include the use of a chaotic map, such as,for example, a one-dimensional chaotic map for independently updatingγ_(l) to γ′_(l) and β_(l) to β′_(l). Examples of one-dimensional chaoticmaps include a logistic map, Kent map, Bernoulli shift map, sine map,ICMIC map, circle map, Chebyshev map, and Gaussian map. Alternatively, atwo-dimensional chaotic map may be used to map a set of γ_(l) and β_(l)to another set of γ′_(l) and β′_(l) by mapping gamma (γ_(l)) and beta(β_(l)) to the first and second dimensions in the chaotic map,respectively. In some embodiments, the variational parameters γ_(l),β_(l) are updated at every iteration (in a discrete manner in timedomain) and thus chaotic maps that are discrete in time domain are used.However, in some embodiments, the variational parameters γ_(l), β_(l)may be updated in a continuous manner in time domain (e.g., based on adifferential equation) and chaotic maps that are continuous in timedomain may be used. However, in general, different chaotic maps thathave different properties in the time domain, space domain, spacedimensions, or number of parameters may be used to perform thecombinatorial optimization process. The chaotic maps (i.e., thefunctions ƒ(γ_(l), p, M, r, s) and g(β_(l), p, M, r, s)) may be tuned(i.e., meta-optimized by adjusting the tuning parameter r) by anotheroptimization method by well-known methods such as meta-evolution,super-optimization, automated parameter calibration, hyper-heuristics,and the like. Properly tuned chaotic maps increase the accuracy of thesolution of the selected combinatorial optimization.

For example, the logistic map may be used to update a set of variationalparameters γ_(l) and β_(l), to γ_(l)′=rγ_(l)(1−γ_(l)) andβ′_(l)=rβ_(l)(1−β_(l)), respectively, during block 818. The tuningparameter r influences performance of the search algorithm (oftenreferred to as the exploration-exploitation tradeoff). Exploration isthe ability to explore various regions in the search space to locate aminimum, preferably a global minimum of the expectation value of themodel Hamiltonian H_(C). Exploitation is the ability to concentrate thesearch around a promising candidate solution in order to locate theminimum of the expectation value of the model Hamiltonian H_(C)precisely. The logistic map is most chaotic when the tuning parameterr=4, allowing one to explore the largest search area to locate a goodminimum (i.e., close to the global minimum) of the expectation value ofthe model Hamiltonian H_(C). The minimum of the expectation value of themodel Hamiltonian H_(C) thus found is an approximate solution to theexact lowest energy of the model Hamiltonian H_(C). In one example, inwhich a simulation using a Gaussian noise source to mimic general noisecharacteristics of a hybrid quantum-classical computing system was used,it was found that when the logistic map with tuning parameter r=4 wasused, the approximation ratio of the approximate solution relative tothe exact lowest energy of the model Hamiltonian H_(C) for a MAXCUTproblem with 7 nodes ranges between 0.68 and 0.72. With the tuningparameter r=2.5, the approximation ratio drops at about 0.58. In thisexample, blocks 806 to 818 are iterated 100,000 times (i.e., the numberof times of the set of variational parameters {right arrow over (γ)},{right arrow over (β)} was updated s=100,000) until the measuredexpectation value of the model Hamiltonian H_(C) sufficiently converges.

The Kent map may also be used to update a set of variational parametersγ_(l) and

${\beta_{l}\mspace{14mu} {to}\mspace{14mu} \gamma_{l}^{\prime}} = \{ {\begin{matrix}{{\frac{\gamma_{l}}{m},}\ } & {0 < \gamma_{l} < m} \\{\frac{1 - \gamma_{l}}{1 - m},} & {m < \gamma_{l} < 1}\end{matrix},{\beta_{l}^{\prime} = \{ {{\begin{matrix}{{\frac{\beta_{l}}{m},}\ } & {0 < \beta_{l} < m} \\{{\frac{1 - \beta_{l}}{1 - m},}\ } & {m < \beta_{l} < 1}\end{matrix}( {{{where}\mspace{14mu} 0} < m < 1} )},} }} $

respectively, during block 818. The Kent map is in the most chaoticregime when the tuning parameter r is the farthest from 0.5. Similar tothe example above, a simulation using a Gaussian noise source was used,in which it was found that when the Kent map with the tuning parameterr=0.99 was used, the approximation ratio ranges between 0.68 and 0.77for a MAXCUT problem with 7 nodes. With the tuning parameter r=0.5, theapproximation ratio was found to drop to about 0.59. In this example,blocks 806 to 818 are iterated 100 times (i.e., the number of times theset of variational parameters {right arrow over (γ)}, {right arrow over(β)} was updated s=100) until the measured expectation value of themodel Hamiltonian H_(C) sufficiently converges. Therefore, in thisexample, by making the selection of the Kent chaotic map versus thelogistic chaotic map provided a significant decrease in the number ofiterations that were required to solve the MAXCUT problem, leading to asignificant decrease in the total processing time due to this selection.One will note that non-Gaussian noise sources can occur in hybridquantum-classical computing systems and the selection of a preferredchaos map can vary depending on the characteristics of the noisy source.Thus, depending on the characteristics of the variational parameters,the chaotic map, and noise exhibited in the hybrid quantum-classicalcomputing system one skilled in the art (or the classical computeritself) may select a preferred chaotic map and tuning parameters.

The functions ƒ(γ_(l), p, M, r, s) and g(β_(l), p, M, r, s) may includedifferent chaotic maps or the same chaotic maps that are differentlytuned in different iterations, depending on the structures of theselected combinatorial optimization problem and/or desired optimizationstrategies.

With the chaotic mapping, optimization of 2p parameters by aconventional classical stochastic optimization algorithm is effectivelyconverted into optimization of a parameters, where the number a is lessthan 2p, resulting in an exponentially smaller variational search space.A smaller variational search space reduces iterations of measuringexpectation values of the model Hamiltonian H_(C), and thus the totalnumber of quantum circuits to run on the quantum processor 106, whichbrings down the time and cost of solving a problem in a hybridquantum-classical computing system.

In block 820, the classical computer 102 will typically output theresults of the variational search to a user interface of the classicalcomputer 102 and/or save the results of the variational search in thememory of the classical computer 102. The results of the variationalsearch will include the measured expectation value of the modelHamiltonian H_(C) in the final iteration corresponding to the minimizedvalue of the objective function C(z)=Σ_(α=1) ^(t)h_(α)C_(α) (z) of theselected combinatorial optimization problem (e.g., a shortest distancefor all of the trips visiting all given cities in a travelling salesmanproblem) and the measurement of the trail state |Ψ_(α)({right arrow over(γ)}, {right arrow over (β)})

in the final iteration corresponding to the solution to the N-bit string(z=z₁ z₂ . . . z_(N)) that provides the minimized value of the objectivefunction C(z)=Σ_(α=1) ^(t)h_(α)C_(α) (z) of the selected combinatorialoptimization problem (e.g., a route of the trips to visit all of thegiven cities that provides the shortest distance for a travellingsalesman).

It should be noted that the particular example embodiments describedabove are just some possible examples of a hybrid quantum-classicalcomputing system according to the present disclosure and do not limitthe possible configurations, specifications, or the like of hybridquantum-classical computing systems according to the present disclosure.For example, a quantum processor within a hybrid quantum-classicalcomputing system is not limited to a group of trapped ions describedabove. For example, a quantum processor may be a superconducting circuitthat includes micrometer-sized loops of superconducting metalinterrupted by a number of Josephson junctions, functioning as qubits(referred to as flux qubits). The junction parameters are engineeredduring fabrication so that a persistent current will flow continuouslywhen an external magnetic flux is applied. As only an integer number offlux quanta are allowed to penetrate in each loop, clockwise orcounter-clockwise persistent currents are developed in the loop tocompensate (screen or enhance) a non-integer external magnetic fluxapplied to the loop. The two states corresponding to the clockwise andcounter-clockwise persistent currents are the lowest energy states;differ only by the relative quantum phase. Higher energy statescorrespond to much larger persistent currents, thus are well separatedenergetically from the lowest two eigenstates. The two lowesteigenstates are used to represent qubit states |0

and |1

. An individual qubit state of each qubit device may be manipulated byapplication of a series of microwave pulses, frequency and duration ofwhich are appropriately adjusted.

The variational search with chaotic maps described herein provides animproved method for optimizing variational parameters by a classicalcomputer within the Quantum Approximate Optimization Algorithm (QAOA)performed on a hybrid quantum-classical computing system. Thus, thefeasibility that a hybrid quantum-classical computing system may allowsolving problems, which are not practically feasible on classicalcomputers, or suggest a considerable speed up with respect to the bestknown classical algorithm even with a noisy intermediate-scale quantumdevice (NISQ) device.

While the foregoing is directed to specific embodiments, other andfurther embodiments may be devised without departing from the basicscope thereof, and the scope thereof is determined by the claims thatfollow.

1. A method of performing computation in a hybrid quantum-classicalcomputing system comprising a classical computer and a quantumprocessor, comprising: selecting, by a classical computer, a problem tobe solved and computing a model Hamiltonian onto which the selectedproblem is mapped; selecting, by the classical computer, a set ofvariational parameters; setting a quantum processor in an initial state,wherein the quantum processor comprises a plurality of qubits;transforming the quantum processor from the initial state to a trialstate based on the computed model Hamiltonian and the selected set ofvariational parameters; measuring an expectation value of the modelHamiltonian on the quantum processor; and determining, by the classicalcomputer, if a difference between the measured expectation value of themodel Hamiltonian is more or less than a predetermined value, whereinthe classical computer either: selects another set of variationalparameters based on a chaotic map if it is determined that thedifference is more than the predetermined value and then: sets thequantum processor in the initial state, transforms the quantum processorfrom the initial state to a new trial state, and measures an expectationvalue of the model Hamiltonian on the quantum processor aftertransforming the quantum processor to the new trial state; or outputsthe measured expectation value of the model Hamiltonian as an optimizedsolution to the selected problem if it is determined that the differenceis less than the predetermined value.
 2. The method according to claim1, wherein the problem to be solved is a combinatorial optimizationproblem.
 3. The method according to claim 1, wherein the chaotic map isa logistic map.
 4. The method according to claim 1, wherein the set ofvariational parameters is selected by the classical computer randomly.5. The method according to claim 1, wherein the quantum processorcomprises a group of trapped ions, each of which has twofrequency-separated states defining a qubit.
 6. The method according toclaim 5, wherein setting the quantum processor in the initial statecomprising setting, by a system controller, each trapped ion in thequantum processor in a superposition of the two frequency-separatedstates.
 7. The method according to claim 5, wherein transforming thequantum processor from the initial state to the trial state comprisesrepeatedly applying, by a system controller, a circuit computed based onthe model Hamiltonian and a mixing circuit to the quantum processor. 8.A hybrid quantum-classical computing system, comprising: a quantumprocessor comprising a group of trapped ions, each of the trapped ionshaving two hyperfine states defining a qubit; one or more lasersconfigured to emit a laser beam, which is provided to trapped ions inthe quantum processor; a classical computer configured to: select aproblem to be solved; compute a model Hamiltonian onto which theselected problem is mapped; and select a set of variational parameters;and a system controller configured to: set the quantum processor in aninitial state; transform the quantum processor from the initial state toa trial state based on the computed model Hamiltonian and the selectedset of variational parameters; and measure an expectation value of themodel Hamiltonian on the quantum processor, wherein the classicalcomputer is further configured to: determine if a difference between themeasure population and a previously measured population of the twofrequency-separated states of each trapped ion in the quantum processoris less than a predetermined value, wherein the classical computerselects another set of variational parameters based on a chaotic map ifit is determined that the difference is more than the predeterminedvalue and then: sets the quantum processor in the initial state,transforms the quantum processor from the initial state to a new trialstate, and measures an expectation value of the model Hamiltonian on thequantum processor after transforming the quantum processor to the newtrial state; or output the measured expectation value of the modelHamiltonian as an optimized solution to the selected problem if it isdetermined that the difference is less than the predetermined value. 9.The hybrid quantum-classical computing system according to claim 8,wherein the problem to be solved is a combinatorial optimizationproblem.
 10. The hybrid quantum-classical computing system according toclaim 8, wherein the chaotic map is logistic map.
 11. The hybridquantum-classical computing system according to claim 8, wherein the setof variational parameters is selected by the classical computerrandomly.
 12. The hybrid quantum-classical computing system according toclaim 8, wherein the quantum processor is set in the initial state bysetting, by a system controller, each trapped ion in the quantumprocessor in a lower state of the two frequency-separated states usingoptical pumping.
 13. The hybrid quantum-classical computing systemaccording to claim 8, wherein the quantum processor is transformed fromthe initial state to the trial state by repeatedly applying, by a systemcontroller, a circuit computed based on the model Hamiltonian and amixing circuit to the quantum processor.
 14. The hybridquantum-classical computing system according to claim 13, wherein acircuit computed based on the model Hamiltonian and a mixing circuit arerepeatedly applied to the quantum processor by applying, by the systemcontroller, a series of laser pulses a series of laser pulses,intensities, durations, and detuning of which are appropriately adjustedby the classical computer on the trapped ions in the quantum processor.15. A hybrid quantum-classical computing system comprising non-volatilememory having a number of instructions stored therein which, whenexecuted by one or more processors, causes the hybrid quantum-classicalcomputing system to perform operations comprising: select a problem tobe solved and computing a model Hamiltonian onto which the selectedproblem is mapped; select a set of variational parameters; set a quantumprocessor in an initial state, wherein the quantum processor comprises aplurality of qubits; transform the quantum processor from the initialstate to a trial state based on the computed model Hamiltonian and theselected set of variational parameters; measure an expectation value ofthe model Hamiltonian on the quantum processor; and determine if adifference between the measured expectation value of the modelHamiltonian is more or less than a predetermined value, wherein thecomputer program instructions further cause the information processingsystem to either: select another set of variational parameters based ona chaotic map if it is determined that the difference is more than thepredetermined value and then: set the quantum processor in the initialstate, transform the quantum processor from the initial state to a newtrial state, and measure an expectation value of the model Hamiltonianon the quantum processor after transforming the quantum processor to thenew trial state; or output the measured population as an optimizedsolution to the selected problem if it is determined that the differenceis less than the predetermined value.
 16. The hybrid quantum-classicalcomputing system according to claim 15, wherein the problem to be solvedis a combinatorial optimization problem.
 17. The hybridquantum-classical computing system according to claim 15 is a logisticmap.
 18. The hybrid quantum-classical computing system according toclaim 15, wherein the quantum processor comprises a group of trappedions, each of which has two frequency-separated states defining a qubit.19. The hybrid quantum-classical computing system according to claim 18,wherein the set of variational parameters is selected by the classicalcomputer randomly.
 20. The hybrid quantum-classical computing systemaccording to claim 18, wherein the quantum processor is set in theinitial state by setting, by a system controller, each trapped ion inthe quantum processor in a superposition of the two frequency-separatedstates.